and a normalized Hilbert Eisenstein series \(G_{\mathbb {F}, {k}}(\tau )\)  (p 20 in [1]}) defined by
\(E_{\mathbb {F}, {k}}(\tau ) &:=& \frac{D^{ \frac{1}{2}-k }(2\pi i)^{tk} }{\Gamma (k)^t} \bigl (\frac{1}{2^t}\zeta _{\mathbb {F}}(1-k)+\sum _{\tiny { \begin{array}{cc}\nu \in \mathfrak {D}^{-1}\\ \nu \succ 0 \end{array}}}\sigma _{k-1}(\nu \mathfrak {D})e^{2\pi i tr(\nu \tau )}\bigr )\\&:=&\frac{D^{ \frac{1}{2}-k }(2\pi i)^{tk} }{\Gamma (k)^t} G_{\mathbb {F}, {k}}(\tau ), \, \sigma _{r}({n})=\sum _{{c}|{n}} \mathcal {N}({c})^{r}.\\\) 
